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Angular momentum L is related to the moment of inertia (MOI) , I by L= Iω

In the body-fixed frame , ie. rotating with the object then ω = 0 and so the angular momentum is zero in the body-fixed frame. Is that correct ?

If i have a thin circular ring then the MOI about the centre is given by I = MR

^{2}where M is the total mass and R is the radius. Is this the MOI in the inertial frame or the body-fixed frame ?

Thanks